Indestructible Man is a 1956 film starring Lon Chaney, Jr. It is freely available online at Google Video:
and is also online at the Internet Archive.
Chaney doesn't speak at all after the first few minutes, but you do get lots of close-ups of his eyes doing a weird twitchy thing. The detective's narration (Max Showalter) overlays the whole movie.
HT: Video WatchBlog via GreenCine